In this section, we use matrices to give a representation of complex numbers. Indeed, consider the set
We will write
Clearly, the set
In particular, we have
for any real numbers a, b, c, and d.
Algebraic Properties of
- 1.
- Addition: For any real numbers a, b, c, and d, we have
Ma,b + Mc,d = Ma+c,b+d.
In other words, if we add two elements of the set, we still get a matrix in
. In particular, we have
-Ma,b = M-a,-b. - 2.
- Multiplication by a number: We have
So a multiplication of an element ofand a number gives a matrix in
.
- 2.
- Multiplication: For any real numbers a, b, c, and d, we have
In other words, we have
Ma,b Mc,d = Mac-bd, ad+bc.
This is an extraordinary formula. It is quite conceivable given the difficult form of the matrix multiplication that, a priori, the product of two elements ofmay not be in
again. But, in this case, it turns out to be true.
The above properties infer to
So, if
In other words, any nonzero element Ma,b of
In order to define the division in
So for the set
Ma, b÷Mc, d = Ma, b×Mc, d-1 = Ma, b×M
,
which implies
Ma, b÷Mc, d = M
, - 
The matrix Ma,-b is called the conjugate of Ma,b. Note that the conjugate of the conjugate of Ma,b is Ma,b itself.
Fundamental Equation. For any Ma,b in
Ma,b = a M1,0 + b M0,1 = a I2 + b M0,1.
Note that
M0,1 M0,1 = M-1,0 = - I2.
Remark. If we introduce an imaginary number i such that i2 = -1, then the matrix Ma,b may be rewritten by
a + bi
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